/*
 *  approx.c:       Approximation of range images with matching pursuit
 *
 *  Written by:     Ullrich Hafner
 *
 *  This file is part of FIASCO (Fractal Image And Sequence COdec)
 *  Copyright (C) 1994-2000 Ullrich Hafner
 */

/*
 *  $Date: 2000/06/14 20:50:51 $
 *  $Author: hafner $
 *  $Revision: 5.1 $
 *  $State: Exp $
 */

#include "config.h"

#include <math.h>

#include "types.h"
#include "macros.h"
#include "error.h"

#include "cwfa.h"
#include "ip.h"
#include "rpf.h"
#include "domain-pool.h"
#include "misc.h"
#include "list.h"
#include "approx.h"
#include "coeff.h"
#include "wfalib.h"

/*****************************************************************************

                 local variables

*****************************************************************************/

typedef struct mp
{
   word_t exclude [MAXEDGES];
   word_t indices [MAXEDGES + 1];
   word_t into [MAXEDGES + 1];
   real_t weight [MAXEDGES];
   real_t matrix_bits;
   real_t weights_bits;
   real_t err;
   real_t costs;
} mp_t;

/*****************************************************************************

                 prototypes

*****************************************************************************/

static void
orthogonalize (unsigned index, unsigned n, unsigned level, real_t min_norm,
           const word_t *domain_blocks, const coding_t *c);
static void
matching_pursuit (mp_t *mp, bool_t full_search, real_t price,
          unsigned max_edges, int y_state, const range_t *range,
          const domain_pool_t *domain_pool, const coeff_t *coeff,
          const wfa_t *wfa, const coding_t *c);

/*****************************************************************************

                public code

*****************************************************************************/

real_t
approximate_range (real_t max_costs, real_t price, int max_edges,
                   int y_state, range_t *range, domain_pool_t *domain_pool,
                   coeff_t *coeff, const wfa_t *wfa, const coding_t *c) {
/*----------------------------------------------------------------------------
  Approximate image block 'range' by matching pursuit. This functions
  calls the matching pursuit algorithm several times (with different
  parameters) in order to find the best approximation. Refer to function
  'matching_pursuit()' for more details about parameters.

  Return value: approximation costs
-----------------------------------------------------------------------------*/
    mp_t mp;

    /*
     *  First approximation attempt: default matching pursuit algorithm.
     */
    mp.exclude [0] = NO_EDGE;
    matching_pursuit(&mp, c->options.full_search, price, max_edges,
                     y_state, range, domain_pool, coeff, wfa, c);

    /*
     *  Next approximation attempt: remove domain block mp->indices [0]
     *  from domain pool (vector with smallest costs) and run the
     *  matching pursuit again.
     */
    if (c->options.second_domain_block) {
        mp_t tmp_mp;

        tmp_mp = mp;  /* initial value */

        tmp_mp.exclude[0] = tmp_mp.indices [0];
        tmp_mp.exclude[1] = NO_EDGE;

        matching_pursuit(&tmp_mp, c->options.full_search, price, max_edges,
                         y_state, range, domain_pool, coeff, wfa, c);
        if (tmp_mp.costs < mp.costs)  /* success */
            mp = tmp_mp;
    }

    /*
     *  Next approximation attempt: check whether some coefficients have
     *  been quantized to zero. Vectors causing the underflow are
     *  removed from the domain pool and then the matching pursuit
     *  algorithm is run again (until underflow doesn't occur anymore).
     */
    if (c->options.check_for_underflow) {
        mp_t tmp_mp;
        int  iteration;

        tmp_mp = mp;  /* initial value */
        iteration = -1;  /* initial value */

        do {
            int i;

            ++iteration;
            tmp_mp.exclude[iteration] = NO_EDGE;

            for (i = 0; isdomain(tmp_mp.indices[i]); ++i) {
                if (tmp_mp.weight [i] == 0) {
                    tmp_mp.exclude[iteration] = tmp_mp.indices [i];
                    break;
                }
            }
            if (isdomain (tmp_mp.exclude [iteration])) {
                /* try again */
                tmp_mp.exclude [iteration + 1] = NO_EDGE;

                matching_pursuit(&tmp_mp, c->options.full_search, price,
                                 max_edges, y_state, range, domain_pool,
                                 coeff, wfa, c);
                if (tmp_mp.costs < mp.costs)     /* success */
                    mp = tmp_mp;
            }
        } while (isdomain (tmp_mp.exclude [iteration])
                 && iteration < MAXEDGES - 1);
    }

    /*
     *  Next approximation attempt: check whether some coefficients have
     *  been quantized to +/- max-value. Vectors causing the overflow are
     *  removed from the domain pool and then the matching pursuit
     *  algorithm is run again (until overflow doesn't occur anymore).
     */
    if (c->options.check_for_overflow) {
        mp_t tmp_mp;
        int  iteration;

        tmp_mp = mp;  /* initial value */
        iteration = -1;  /* initial value */

        do {
            int i;

            ++iteration;
            tmp_mp.exclude[iteration] = NO_EDGE;

            for (i = 0; isdomain (tmp_mp.indices [i]); ++i) {
                rpf_t * const rpf =
                    tmp_mp.indices [i] ? coeff->rpf : coeff->dc_rpf;

                if (tmp_mp.weight [i] == btor (rtob (200, rpf), rpf)
                    || tmp_mp.weight [i] == btor (rtob (-200, rpf), rpf)) {
                    tmp_mp.exclude [iteration] = tmp_mp.indices [i];
                    break;
                }
            }

            if (isdomain(tmp_mp.exclude[iteration])) {
                /* try again */
                tmp_mp.exclude[iteration + 1] = NO_EDGE;

                matching_pursuit(&tmp_mp, c->options.full_search, price,
                                 max_edges, y_state, range, domain_pool,
                                 coeff, wfa, c);
                if (tmp_mp.costs < mp.costs)   /* success */
                    mp = tmp_mp;
            }
        } while (isdomain (tmp_mp.exclude [iteration])
                 && iteration < MAXEDGES - 1);
    }

    /*
     *  Finally, check whether the best approximation has costs
     *  smaller than 'max_costs'.
     */
    if (mp.costs < max_costs) {
        int    edge;
        int    new_index, old_index;

        new_index = 0;
        for (old_index = 0; isdomain (mp.indices[old_index]); ++old_index) {
            if (mp.weight [old_index] != 0) {
                mp.indices [new_index] = mp.indices [old_index];
                mp.into    [new_index] = mp.into    [old_index];
                mp.weight  [new_index] = mp.weight  [old_index];
                ++new_index;
            }
        }
        mp.indices [new_index] = NO_EDGE;
        mp.into    [new_index] = NO_EDGE;

        /*
         *  Update of probability models
         */
        {
            word_t * const domain_blocks =
                domain_pool->generate(range->level, y_state,
                                      wfa,
                                      domain_pool->model);
            domain_pool->update(domain_blocks, mp.indices,
                                range->level, y_state, wfa,
                                domain_pool->model);
            coeff->update (mp.weight, mp.into, range->level, coeff);

            Free(domain_blocks);
        }

        for (edge = 0; isedge (mp.indices [edge]); ++edge) {
            range->into   [edge] = mp.into   [edge];
            range->weight [edge] = mp.weight [edge];
        }
        range->into [edge]  = NO_EDGE;
        range->matrix_bits  = mp.matrix_bits;
        range->weights_bits = mp.weights_bits;
        range->err          = mp.err;
    } else {
        range->into [0] = NO_EDGE;
        mp.costs        = MAXCOSTS;
    }

    return mp.costs;
}



/*****************************************************************************

                 local variables

*****************************************************************************/

static real_t norm_ortho_vector [MAXSTATES];
/*
 *  Square-norm of the i-th vector of the orthogonal basis (OB)
 *  ||o_i||^2; i = 0, ... ,n
 */
static real_t ip_image_ortho_vector [MAXEDGES];
/*
 *  Inner product between the i-th vector of the OB and the given range:
 *  <b, o_i>; i = 0, ... ,n
 */
static real_t ip_domain_ortho_vector [MAXSTATES][MAXEDGES];
/*
 *  Inner product between the i-th vector of the OB and the image of domain j:
 *  <s_j, o_i>; j = 0, ... , wfa->states; i = 0, ... ,n,
 */
static real_t rem_denominator [MAXSTATES];
static real_t rem_numerator [MAXSTATES];
/*
 *  At step n of the orthogonalization the comparative value
 *  (numerator_i / denominator_i):= <b, o_n>^2 / ||o_n|| ,
 *  is computed for every domain i,
 *  where o_n := s_i - \sum(k = 0, ... , n-1) {(<s_i, o_k> / ||o_k||^2) o_k}
 *  To avoid computing the same values over and over again,
 *  the constant (remaining) parts of every domain are
 *  stored in 'rem_numerator' and 'rem_denominator' separately
 */
static bool_t used [MAXSTATES];
/*
 *  Shows whether a domain image was already used in a
 *  linear combination (YES) or not (NO)
 */

/*****************************************************************************

                private code

*****************************************************************************/

static void
matching_pursuit (mp_t *mp, bool_t full_search, real_t price,
                  unsigned max_edges, int y_state, const range_t *range,
                  const domain_pool_t *domain_pool, const coeff_t *coeff,
                  const wfa_t *wfa, const coding_t *c)
/*
 *  Find an approximation of the current 'range' with a linear
 *  combination of vectors of the 'domain_pool'. The linear
 *  combination is generated step by step with the matching pursuit
 *  algorithm.  If flag 'full_search' is set then compute complete set
 *  of linear combinations with n = {0, ..., 'max_edges'} vectors and
 *  return the best one. Otherwise abort the computation as soon as
 *  costs (LC (n + 1)) exceed costs ( LC (n)) and return the
 *  sub-optimal solution.  'price' is the langrange multiplier
 *  weighting rate and distortion.  'band' is the current color band
 *  and 'y_state' the corresponding state in the Y component at same
 *  pixel position.  'domain_pool' gives the set of available vectors,
 *  and 'coeff' the model for the linear factors. The number of
 *  elements in the linear combination is limited by 'max_edges'. In
 *  'mp', vectors may be specified which should be excluded during the
 *  approximation.
 *
 *  No return value.
 *
 *  Side effects:
 *  vectors, factors, rate, distortion and costs are stored in 'mp'
 */
{
    unsigned  n;         /* current vector of the OB */
    int       index;         /* best fitting domain image */
    unsigned  domain;        /* counter */
    real_t    norm;          /* norm of range image */
    real_t    additional_bits;   /* bits for mc, nd, and tree */
    word_t   *domain_blocks;     /* current set of domain images */
    const real_t  min_norm = 2e-3;   /* lower bound of norm */
    unsigned      best_n   = 0;
    unsigned  size     = size_of_level (range->level);

    /*
     *  Initialize domain pool and inner product arrays
     */
    domain_blocks = domain_pool->generate (range->level, y_state, wfa,
                                           domain_pool->model);
    for (domain = 0; domain_blocks [domain] >= 0; domain++)
    {
        used [domain] = NO;
        rem_denominator [domain] =     /* norm of domain */
            get_ip_state_state (domain_blocks [domain], domain_blocks [domain],
                                range->level, c);
        if (rem_denominator [domain] / size < min_norm)
            used [domain] = YES;       /* don't use domains with small norm */
        else
            rem_numerator [domain]     /* inner product <s_domain, b> */
                = get_ip_image_state (range->image, range->address,
                                      range->level, domain_blocks [domain], c);
        if (!used [domain] && fabs (rem_numerator [domain]) < min_norm)
            used [domain] = YES;
    }

    /*
     *  Exclude all domain blocks given in array 'mp->exclude'
     */
    for (n = 0; isdomain (mp->exclude [n]); n++)
        used [mp->exclude [n]] = YES;

    /*
     *  Compute the approximation costs if 'range' is approximated with
     *  no linear combination, i.e. the error is equal to the square
     *  of the image norm and the size of the automaton is determined by
     *  storing only zero elements in the current matrix row
     */
    for (norm = 0, n = 0; n < size; n++)
        norm += square (c->pixels [range->address * size + n]);

    additional_bits = range->tree_bits + range->mv_tree_bits
        + range->mv_coord_bits + range->nd_tree_bits
        + range->nd_weights_bits;

    mp->err          = norm;
    mp->weights_bits = 0;
    mp->matrix_bits  = domain_pool->bits (domain_blocks, NULL, range->level,
                                          y_state, wfa, domain_pool->model);
    mp->costs        = (mp->matrix_bits + mp->weights_bits
                        + additional_bits) * price + mp->err;

    n = 0;
    do
    {
        /*
         *  Current approximation is: b = d_0 o_0 + ... + d_(n-1) o_(n-1)
         *  with corresponding costs 'range->err + range->bits * p'.
         *  For all remaining state images s_i (used[s_i] == NO) set
         *  o_n : = s_i - \sum(k = 0, ... , n-1) {(<s_i, o_k> / ||o_k||^2) o_k}
         *  and try to beat current costs.
         *  Choose that vector for the next orthogonalization step,
         *  which has minimal costs: s_index.
         *  (No progress is indicated by index == -1)
         */

        real_t min_matrix_bits  = 0;
        real_t min_weights_bits = 0;
        real_t min_error        = 0;
        real_t min_weight [MAXEDGES];
        real_t min_costs = full_search ? MAXCOSTS : mp->costs;

        for (index = -1, domain = 0; domain_blocks [domain] >= 0; domain++)
            if (!used [domain])
            {
                real_t    matrix_bits, weights_bits;
                /*
                 *  To speed up the search through the domain images,
                 *  the costs of using domain image 'domain' as next vector
                 *  can be approximated in a first step:
                 *  improvement of image quality
                 *  <= square (rem_numerator[domain]) / rem_denominator[domain]
                 */
                {
                    word_t   vectors [MAXEDGES + 1];
                    word_t   states [MAXEDGES + 1];
                    real_t   weights [MAXEDGES + 1];
                    unsigned i, k;

                    for (i = 0, k = 0; k < n; k++)
                        if (mp->weight [k] != 0)
                        {
                            vectors [i] = mp->indices [k];
                            states [i]  = domain_blocks [vectors [i]];
                            weights [i] = mp->weight [k];
                            i++;
                        }
                    vectors [i]     = domain;
                    states [i]      = domain_blocks [domain];
                    weights [i]     = 0.5;
                    vectors [i + 1] = -1;
                    states [i + 1]  = -1;

                    weights_bits = coeff->bits (weights, states, range->level,
                                                coeff);
                    matrix_bits = domain_pool->bits (domain_blocks, vectors,
                                                     range->level, y_state,
                                                     wfa, domain_pool->model);
                }
                if (((matrix_bits + weights_bits + additional_bits) * price +
                     mp->err -
                     square (rem_numerator[domain]) / rem_denominator[domain])
                    < min_costs)
                {
                    /*
                     *  1.) Compute the weights (linear factors) c_i of the
                     *  linear combination
                     *  b = c_0 v_0 + ... + c_(n-1) v_(n-1) + c_n v_'domain'
                     *  Use backward substitution to obtain c_i from the linear
                     *  factors of the lin. comb. b = d_0 o_0 + ... + d_n o_n
                     *  of the corresponding orthogonal vectors {o_0, ..., o_n}.
                     *  Vector o_n of the orthogonal basis is obtained by using
                     *  vector 'v_domain' in step n of the Gram Schmidt
                     *  orthogonalization (see above for definition of o_n).
                     *  Recursive formula for the coefficients c_i:
                     *  c_n := <b, o_n> / ||o_n||^2
                     *  for i = n - 1, ... , 0:
                     *  c_i := <b, o_i> / ||o_i||^2 +
                     *          \sum (k = i + 1, ... , n){ c_k <v_k, o_i>
                     *                   / ||o_i||^2 }
                     *  2.) Because linear factors are stored with reduced
                     *  precision factor c_i is rounded with the given
                     *  precision in step i of the recursive formula.
                     */

                    unsigned k;      /* counter */
                    int      l;      /* counter */
                    real_t   m_bits;     /* number of matrix bits to store */
                    real_t   w_bits;     /* number of weights bits to store */
                    real_t   r [MAXEDGES];   /* rounded linear factors */
                    real_t   f [MAXEDGES];   /* linear factors */
                    int      v [MAXEDGES];   /* mapping of domains to vectors*/
                    real_t   costs;      /* current approximation costs */
                    real_t   m_err;      /* current approximation error */

                    f [n] = rem_numerator [domain] / rem_denominator [domain];
                    v [n] = domain;      /* corresponding mapping */
                    for (k = 0; k < n; k++)
                    {
                        f[k] = ip_image_ortho_vector[k] / norm_ortho_vector[k];
                        v [k] = mp->indices [k];
                    }

                    for (l = n; l >= 0; --l) {
                        rpf_t * const rpf = domain_blocks[v[l]]
                            ? coeff->rpf : coeff->dc_rpf;

                        unsigned int k;

                        r[l] = f[l] = btor(rtob(f[l], rpf), rpf);

                        {
                            real_t const fl = f[l];

                            for (k = 0; k < l; ++k) {
                                f[k] -= fl * ip_domain_ortho_vector[v[l]][k]
                                    / norm_ortho_vector[k];
                            }
                        }
                    }

                    /*
                     *  Compute the number of output bits of the linear
                     *  combination and store the weights with reduced
                     *  precision. The resulting linear combination is
                     *  b = r_0 v_0 + ... + r_(n-1) v_(n-1) + r_n v_'domain'
                     */
                    {
                        word_t vectors [MAXEDGES + 1];
                        word_t states [MAXEDGES + 1];
                        real_t weights [MAXEDGES + 1];
                        int    i;

                        for (i = 0, k = 0; k <= n; k++)
                            if (f [k] != 0)
                            {
                                vectors [i] = v [k];
                                states [i]  = domain_blocks [v [k]];
                                weights [i] = f [k];
                                i++;
                            }
                        vectors [i] = -1;
                        states [i]  = -1;

                        w_bits =
                            coeff->bits(weights, states, range->level, coeff);
                        m_bits = domain_pool->bits (domain_blocks, vectors,
                                                    range->level, y_state,
                                                    wfa, domain_pool->model);
                    }

                    /*
                     *  To compute the approximation error, the corresponding
                     *  linear factors of the linear combination
                     *  b = r_0 o_0 + ... + r_(n-1) o_(n-1) + r_n o_'domain'
                     *  with orthogonal vectors must be computed with following
                     *  formula:
                     *  r_i := r_i +
                     *          \sum (k = i + 1, ... , n) { r_k <v_k, o_i>
                     *                   / ||o_i||^2 }
                     */
                    for (l = 0; (unsigned) l <= n; l++)
                    {
                        /*
                         *  compute <v_n, o_n>
                         */
                        real_t a;

                        a = get_ip_state_state (domain_blocks [v [l]],
                                                domain_blocks [domain],
                                                range->level, c);
                        for (k = 0; k < n; k++)
                            a -= ip_domain_ortho_vector [v [l]][k]
                                / norm_ortho_vector [k]
                                * ip_domain_ortho_vector [domain][k];
                        ip_domain_ortho_vector [v [l]][n] = a;
                    }
                    norm_ortho_vector [n]     = rem_denominator [domain];
                    ip_image_ortho_vector [n] = rem_numerator [domain];

                    for (k = 0; k <= n; k++)
                        for (l = k + 1; (unsigned) l <= n; l++)
                            r [k] += ip_domain_ortho_vector [v [l]][k] * r [l]
                                / norm_ortho_vector [k];
                    /*
                     *  Compute approximation error:
                     *  error := ||b||^2 +
                     *  \sum (k = 0, ... , n){r_k^2 ||o_k||^2 - 2 r_k <b, o_k>}
                     */
                    m_err = norm;
                    for (k = 0; k <= n; k++)
                        m_err += square (r [k]) * norm_ortho_vector [k]
                            - 2 * r [k] * ip_image_ortho_vector [k];
                    if (m_err < 0)       /* TODO: return MAXCOSTS */
                        warning ("Negative image norm: %f"
                                 " (current domain: %d, level = %d)",
                                 (double) m_err, domain, range->level);

                    costs = (m_bits + w_bits + additional_bits) * price + m_err;
                    if (costs < min_costs)   /* found a better approximation */
                    {
                        index            = domain;
                        min_costs        = costs;
                        min_matrix_bits  = m_bits;
                        min_weights_bits = w_bits;
                        min_error        = m_err;
                        for (k = 0; k <= n; k++)
                            min_weight [k] = f [k];
                    }
                }
            }

        if (index >= 0)           /* found a better approximation */
        {
            if (min_costs < mp->costs)
            {
                unsigned k;

                mp->costs        = min_costs;
                mp->err          = min_error;
                mp->matrix_bits  = min_matrix_bits;
                mp->weights_bits = min_weights_bits;

                for (k = 0; k <= n; k++)
                    mp->weight [k] = min_weight [k];

                best_n = n + 1;
            }

            mp->indices [n] = index;
            mp->into [n]    = domain_blocks [index];

            used [index] = YES;

            /*
             *  Gram-Schmidt orthogonalization step n
             */
            orthogonalize (index, n, range->level, min_norm, domain_blocks, c);
            n++;
        }
    }
    while (n < max_edges && index >= 0);

    mp->indices [best_n] = NO_EDGE;

    mp->costs = (mp->matrix_bits + mp->weights_bits + additional_bits) * price
        + mp->err;

    Free (domain_blocks);
}

static void
orthogonalize (unsigned index, unsigned n, unsigned level, real_t min_norm,
           const word_t *domain_blocks, const coding_t *c)
/*
 *  Step 'n' of the Gram-Schmidt orthogonalization procedure:
 *  vector 'index' is orthogonalized with respect to the set
 *  {u_[0], ... , u_['n' - 1]}. The size of the image blocks is given by
 *  'level'. If the denominator gets smaller than 'min_norm' then
 *  the corresponding domain is excluded from the list of available
 *  domain blocks.
 *
 *  No return value.
 *
 *  Side effects:
 *  The remainder values (numerator and denominator) of
 *  all 'domain_blocks' are updated.
 */
{
   unsigned domain;

   ip_image_ortho_vector [n] = rem_numerator [index];
   norm_ortho_vector [n]     = rem_denominator [index];

   /*
    *  Compute inner products between all domain images and
    *  vector n of the orthogonal basis:
    *  for (i = 0, ... , wfa->states)
    *  <s_i, o_n> := <s_i, v_n> -
    *      \sum (k = 0, ... , n - 1){ <v_n, o_k> <s_i, o_k> / ||o_k||^2}
    *  Moreover the denominator and numerator parts of the comparative
    *  value are updated.
    */
   for (domain = 0; domain_blocks [domain] >= 0; domain++)
      if (!used [domain])
      {
     unsigned k;
     real_t   tmp = get_ip_state_state (domain_blocks [index],
                        domain_blocks [domain], level, c);

     for (k = 0; k < n; k++)
        tmp -= ip_domain_ortho_vector [domain][k] / norm_ortho_vector [k]
           * ip_domain_ortho_vector [index][k];
     ip_domain_ortho_vector [domain][n] = tmp;
     rem_denominator [domain] -= square (tmp) / norm_ortho_vector [n];
     rem_numerator [domain]   -= ip_image_ortho_vector [n]
                     / norm_ortho_vector [n]
                     * ip_domain_ortho_vector [domain][n] ;

     /*
      *  Exclude vectors with small denominator
      */
     if (!used [domain])
        if (rem_denominator [domain] / size_of_level (level) < min_norm)
           used [domain] = YES;
      }
}



